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1
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2
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- Product rule P(aÙb) = P(a |
b) P(b) = P(b | a) P(a)
- Þ Bayes' rule: P(a | b) =
P(b | a) P(a) / P(b)
- or in distribution form
- P(Y|X) = P(X|Y) P(Y) / P(X) = αP(X|Y) P(Y)
- Useful for assessing diagnostic probability from causal probability:
- P(Cause|Effect) = P(Effect|Cause) P(Cause) / P(Effect)
- E.g., let M be meningitis, S be stiff neck:
- P(m|s) = P(s|m) P(m) / P(s) = 0.5 × 0.0002 / 0.05 = 0.0002
- Note: posterior probability of meningitis still very small!
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3
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- P(Cavity | toothache Ù catch)
- = α · P(toothache Ù
catch | Cavity) P(Cavity)
- = α · P(toothache | Cavity) P(catch | Cavity) P(Cavity)
- This is an example of a naïve Bayes model:
- P(Cause,Effect1, … ,Effectn) = P(Cause) πiP(Effecti|Cause)
- Total number of parameters is linear in n
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4
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- Calculate most probable function value
- Vmap = argmax P(vj| a1,a2,
… , an)
- = argmax P(a1,a2,
… , an| vj) P(vj)
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P(a1,a2, … , an)
- = argmax P(a1,a2,
… , an| vj) P(vj)
- Naïve assumption: P(a1,a2, … , an) =
P(a1)P(a2) … P(an)
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5
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- NaïveBayesLearn(examples)
For each target value vj
P’(vj) ←
estimate P(vj)
For each attribute value
ai of each attribute a
P’(ai|vj)
← estimate P(ai|vj)
- ClassfyingNewInstance(x)
vnb= argmax P’(vj) Π P’(ai|vj)
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6
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- (due to MIT’s open coursework slides)
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7
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- (due to MIT’s open coursework slides)
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Notes